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\begin{document}
\pagenumbering{gobble}
\begin{center}
\textbf{The Division Algorithm and Proof Writing}
\end{center}

We need to revisit the exercises from the Week 4 homework and hopefully gain a better knowledge of proof writing.

\begin{center}
\fbox{\parbox{5.5in}{Goals:
	\begin{itemize}
	\item Revisit the definitions of even and odd and the division algorithm.
	\item Formulate a formal proof that an odd integer has the form $2k+1$.
	\item Apply the division algorithm in another setting.
	\end{itemize}
}}
\end{center}

\begin{definition}[Even] An integer $n$ is \textit{even} if $n=2k$ for some $k \in \mathbb{Z}$.
\end{definition}

\begin{definition}[Odd] An integer $n$ is \textit{odd} if it is not even.
\end{definition}

\noindent \textbf{Claim 1:} If an integer $n$ is odd, then $n=2k+1$ for some $k \in \Z$.

\begin{enumerate}
\item Write a formal proof of Claim 1.

\end{enumerate}

\newpage

\noindent \textbf{Claim 2:}  For any integer $n$, either $n$, $n+1$, or $n+2$ is divisible by 3.

\begin{enumerate}
\setcounter{enumi}{1}
\item Write a formal proof of Claim 2.
\end{enumerate}
\end{document}




